## Diploma Thesis

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Bulk-boundary correspondence in non-equilibrium dynamics of one-dimensional topological insulators
(2017)

Dynamical phase transitions (DPT) are receiving a rising interest. They are known to behave analogously to
equilibrium phase transitions (EPT) to a large extend. However, it is easy to see that DPT can occur in finite
systems, while EPT are only possible in the thermodynamic limit. So far it is not clear how far the analogy of
DPT and EPT goes. It was suggested, that there is a relation between topological phase transitions (TPT)
and DPT, but many open questions remain.
Typically, to study DPT, the Loschmidt echo (LE) after a quench is investigated, where DPT are visible as
singularities. For one-dimensional systems, each singularity is connected to a certain critical time scale, which
is given by the dispersion in the chain.
In topological free-fermion models with winding numbers 0 or 1, only the LE in periodic boundary conditions
(PBC) has been investigated. In open boundary conditions (OBC), these models are characterized by symmetry
protected edge modes in the topologically non-trivial phase. It is completely unclear how these modes affect
DPT. We investigate systems with PBC governed by multiple time scales with a Z topological invariant. In
OBC, we provide numerical evidence for the presence of bulk-boundary correspondence in DPT in quenches
across a TPT.

This diploma thesis sets out to analyse the applicability of the instrument ’European Grouping of
Territorial Cooperation (EGTC)’ in the transnational and interregional non-contiguous cooperation.
EGTCs that are applied in spatially non-contiguous cooperations are called ’Network-EGTCs’. As
no scientific research about network-EGTCs has been made so far, this diploma thesis fills this
research gap.
As a basis for the analysis, a literature review on the instrument EGTC in general and its historic
background was conducted. In addition the scientific literature has been searched for characteristics
of non-contiguous cooperations and different stakeholders were interviewed for their
estimations about network-EGTCs. The so far existing and planned network-EGTCs have been
explored. Out of these network-EGTCs two case studies – the E.G.T.C. Amphictyony and the
planned CETC-EGTC – have been examined in depth. Their characteristics have further been
compared with the information about EGTCs and non contiguous-cooperations in general.
It was found out that network-EGTCs show advantages from ordinary non-contiguous cooperations.
Additionally, it was discovered that network-EGTCs do not differ in their character as much
as it had been expected from EGTCs established in the cross-border cooperation. This applies
also to the establishment process as well as to the fulfilment of the instrument’s potentials. In
general all EGTCs show discrepancies between planning and practice. Only a few differences
have been discovered. Contrary to expectation network-EGTCs show only certain disadvantages
but also advantages compared to EGTCs in the cross-border cooperation.
This thesis delivers evidence that EGTCs are applicable in the transnational and interregional
cooperation when certain preconditions are fulfilled. Then they can contribute to a successful
transnational and interregional cooperation.
Recommendations were given to territorial non-contiguous cooperations that are considering to
establish an EGTC.
It is expected that more network-EGTCs will be established in the future due to the higher experience
and knowledge about network-EGTCs.

Tropical geometry is a very new mathematical domain. The appearance of
tropical geometry was motivated by its deep relations to other mathematical
branches. These include algebraic geometry, symplectic geometry, complex
analysis, combinatorics and mathematical biology.
In this work we see some more relations between algebraic geometry and
tropical geometry. Our aim is to prove a one-to-one correspondence between
the divisor classes on the moduli space of n-pointed rational stable curves
and the divisors of the moduli space of n-pointed abstract tropical curves.
Thus we state some results of the algebraic case first. In algebraic geometry
these moduli spaces are well understood. In particular, the group of divisor
classes is calculated by S. Keel. We recall the needed results in chapter one.
For the proof of the correspondence we use some results of toric geometry.
Further we want to show an equality of the Chow groups of a special toric
variety and the algebraic moduli space. Thus we state some results of the
toric geometry as well.
This thesis tries to discover some connection between algebraic and tropical
geometry. Thus we also need the corresponding tropical objects to the
algebraic objects. Therefore we give some necessary definitions such as fan,
tropical fan, morphisms between tropical fans, divisors or the topical moduli
space of all n-marked tropical curves. Since we need it, we show that the
tropical moduli space can be embedded as a tropical fan.
After this preparatory work we prove that the group of divisor classes in
v
classical algebraic geometry has it equivalence in tropical geometry. For this
it is useful to give a map from the group of divisor classes of the algebraic
moduli space to the group of divisors of the tropical moduli space. Our aim is
to prove the bijectivity of this map in chapter three. On the way we discover
a deep connection between the algebraic moduli space and the toric variety
given by the tropical fan of the tropical moduli space.

In this work a 3-dimensional contact elasticity problem for a thin fiber and a rigid foundation is studied. We describe the contact condition by a linear Robin-condition (by meaning of the penalized and linearized non-penetration and friction conditions).
The dimension of the problem is reduced by an asymptotic approach. Scaling the Robin parameters appropriately we obtain a recurrent chain of Neumann type boundary value problems which are considered only in the microscopic scale. The problem for the leading term is a homogeneous Neumann problem, hence the leading term depends only on the slow variable. This motivates the choice of a multiplicative ansatz in the asymptotic expansion.
The theoretical results are illustrated with numerical examples performed with a commercial finite-element software-tool.

In this thesis we present the implementation of libraries center.lib and perron.lib for the non-commutative extension Plural of the Computer Algebra System Singular. The library center.lib was designed for the computation of elements of the centralizer of a set of elements and the center of a non-commutative polynomial algebra. It also provides solutions to related problems. The library perron.lib contains a procedure for the computation of relations between a set of pairwise commuting polynomials. The thesis comprises the theory behind the libraries, aspects of the implementation and some applications of the developed algorithms. Moreover, we provide extensive benchmarks for the computation of elements of the center. Some of our examples were never computed before.

The scope of this diploma thesis is to examine the four generations of asset pricing models and the corresponding volatility dynamics which have been devepoled so far. We proceed as follows: In chapter 1 we give a short repetition of the Black-Scholes first generation model which assumes a constant volatility and we show that volatility should not be modeled as constant by examining statistical data and introducing the notion of implied volatility. In chapter 2, we examine the simplest models that are able to produce smiles or skews - local volatility models. These are called second generation models. Local volatility models model the volatility as a function of the stock price and time. We start with the work of Dupire, show how local volatility models can be calibrated and end with a detailed discussion of the constant elasticity of volatility model. Chapter 3 focuses on the Heston model which represents the class of the stochastic volatility models, which assume that the volatility itself is driven by a stochastic process. These are called third generation models. We introduce the model structure, derive a partial differential pricing equation, give a closed-form solution for European calls by solving this equation and explain how the model is calibrated. The last part of chapter 3 then deals with the limits and the mis-specifications of the Heston model, in particular for recent exotic options like reverse cliquets, Accumulators or Napoleons. In chapter 4 we then introduce the Bergomi forward variance model which is called fourth generation model as a consequence of the limits of the Heston model explained in chapter 3. The Bergomi model is a stochastic local volatility model - the spot price is modeled as a constant elasticity of volatility diffusion and its volatility parameters are functions of the so called forward variances which are specified as stochastic processes. We start with the model specification, derive a partial differential pricing equation, show how the model has to be calibrated and end with pricing examples and a concluding discussion.

This work is concerned with dynamic flow problems, especially maximal dynamic flows and earliest arrival flows - also called universally maximal flows. First of all, a survey of known results about existence, computation and approximation of earliest arrival flows is given. For the special case of series-parallel graphs a polynomial algorithm for computing maximal dynamic flows is presented and this maximal dynamic flow is proven to be an earliest arrival flow.

In an undirected graph G we associate costs and weights to each edge. The weight-constrained minimum spanning tree problem is to find a spanning tree of total edge weight at most a given value W and minimum total costs under this restriction. In this thesis a literature overview on this NP-hard problem, theoretical properties concerning the convex hull and the Lagrangian relaxation are given. We present also some in- and exclusion-test for this problem. We apply a ranking algorithm and the method of approximation through decomposition to our problem and design also a new branch and bound scheme. The numerical results show that this new solution approach performs better than the existing algorithms.

A hub location problem consists of locating p hubs in a network in order to collect and consolidate flow between node pairs. This thesis deals with the uncapacitated single allocation p-hub center problem (USApHCP) as a special type of hub location problem with min max objective function. Using the so-called radius formulation of the problem, the dimension of the polyhedron of USApHCP is derived. The formulation constraints are investigated to find out which of these define facets. Then, three new classes of facet-defining inequalities are derived. Finally, efficient procedures to separate facets in a branch-and-cut algorithm are proposed. The polyhedral analysis of USApHCP is based on a tight relation to the uncapacitated facility location problem (UFL). Hence, many results stated in this thesis also hold for UFL.